Find the time period of mass M when displaced from its equilibrium position and then reelased for the system shown in figure.

Explanation

Let k be the spring constant of the spring. When mass M is suspended from the given system as shown in figure.let l be the length through which mass M moves down before it comes to rest. In this situation, both the spring and string will be stretched by length l. Since string is inextensible, so spring is stretched by length 2l. The tension along the string is the same. So, restoring force is
`F=-2kxx(2l)=-4kl` ...(i)
In equilibrium position,this restoring balances weigth of the mass. i.e.,
`Mg=4kl`
If mass M is pulled doen through small distance y, then restoring force `F^(')` is given by
`F^(')=-2k(2l+2y)`
Effective restoring force on mass M is given by
`f=F^(')-F=-2(2l+2y)-(-4kl)=-4ky` ...(iii)
As `fpropy` and f is directed towards equilibrium position. hence if mass is left free, it will execute linear SHM. From (ii), spring factor`=4k`.
Inertia factor `=M`
As time period, `T=2pisqrt((i n ertia fact o r)/(spri ng fact o r))=2pisqrt((M)/(4k))`